Discrete — Definition, Meaning & Examples
Discrete
A set with elements that are disconnected. The set of integers is discrete. The set of real numbers is not discrete; it is continuous.
Formally, a set of numbers is discrete if each number in the set is contained in a neighborhood that contains no other elements of the set.
Worked Example
Problem: Determine whether the set S = {2, 4, 6, 8, 10} is discrete or continuous.
Step 1: Pick any element of S and check whether you can find a small interval around it that contains no other elements of S. Take the element 4.
Step 2: Consider the open interval (3, 5). This interval contains 4 but no other element of S, since the nearest elements are 2 and 6.
4∈(3,5)andS∩(3,5)={4}
Step 3: Repeat for every element: 2 is isolated in (1, 3), 6 in (5, 7), 8 in (7, 9), and 10 in (9, 11). Each element sits alone in some neighborhood.
Step 4: Because every element of S has a neighborhood containing no other elements of S, the formal definition is satisfied.
Answer: S = {2, 4, 6, 8, 10} is a discrete set.
Another Example
Problem: Is the set of all real numbers between 0 and 1 discrete?
Step 1: Pick any element, say 0.5, and try to find an open interval around it that contains no other elements of the set.
Step 2: No matter how small you make the interval — (0.499, 0.501), (0.4999, 0.5001), etc. — it always contains infinitely many other real numbers from the set.
Step 3: Since you cannot isolate any element in a neighborhood, the set is not discrete. It is continuous.
Answer: The set of all real numbers between 0 and 1 is not discrete; it is continuous.
Frequently Asked Questions
What is the difference between discrete and continuous data?
Discrete data can only take specific, separated values (like the number of students in a class: 25, 26, 27…). Continuous data can take any value within a range, including decimals and fractions (like height: 165.3 cm, 165.31 cm, etc.). The key test is whether there are gaps between possible values.
Can a discrete set be infinite?
Yes. The set of all integers {..., −2, −1, 0, 1, 2, ...} is both infinite and discrete. Having infinitely many elements does not prevent each element from being isolated from the others. What matters is that gaps exist between consecutive elements, not how many elements there are.
Discrete vs. Continuous
A discrete set has isolated points with gaps between them — you can list or count the elements one by one. A continuous set has no gaps; between any two elements there are infinitely many others. The integers are discrete because there is no integer between 3 and 4. The real numbers are continuous because between any two reals (say 3.1 and 3.2) you can always find another (3.15). In data analysis, counts (number of pets) are discrete, while measurements (weight, temperature) are continuous.
Why It Matters
The discrete-versus-continuous distinction shapes how you do math with a set. In statistics, discrete data is handled with probability mass functions and summation, while continuous data uses probability density functions and integration. Understanding whether your data or variable is discrete also determines which graphs (bar charts vs. histograms), formulas, and statistical tests are appropriate.
Common Mistakes
Mistake: Confusing "discrete" with "discreet."
Correction: "Discrete" (with the final 'e' before 't') is the mathematical term meaning separate or distinct. "Discreet" means careful or tactful — it has nothing to do with math.
Mistake: Assuming that any set with a finite number of elements is automatically discrete, or that infinite sets cannot be discrete.
Correction: Finiteness is sufficient for discreteness, but not necessary. Infinite sets like the integers or even the set of all rational numbers with denominator 1 are also discrete. The defining property is isolation of each point, not the total count of elements.
Related Terms
- Set — A collection whose nature may be discrete
- Element of a Set — Individual member that may be isolated
- Integers — The most common example of a discrete set
- Real Numbers — A continuous set, contrasted with discrete
- Continuous — The opposite of discrete — no gaps
- Neighborhood — Used in the formal definition of discrete
- Natural Numbers — Another standard example of a discrete set